Regarding exorcisms, one aspect that is not addressed is sometimes external manifestations of demonic activity/presence not explainable (or seemingly not explainable) by biochemical processes.
Now, one can dismiss such claims out of hand if one considers a priori that such occurrences are impossible (i.e. that the super-natural does not exist), and so are necessarily explainable as deceptions, misperceptions, or yet-to-be understood natural phenomena. That's fine, but then care should be taken that one does not vacillate between that position and claiming that no one has ever shown any evidence of supernatural phenomena. Anyway, the latter point is a bit of a nit, but I've seen armchair skeptics make the mistake often enough that it bears pointing out.
In my own life, I have on several occasions observed or experienced some phenomena which seemed to defy laws of nature. With respect to the demonic, I can relate one such experience, from which I must necessarily omit certain details so as to not betray anyone's privacy:
Several years ago, I met someone whom I will refer to as "Jim". Jim suffered from serious mental illness and had been unable to develop an awareness that he had a problem for which medical treatment was necessary (and readily available). I tried to be a good friend, but Jim's life was upside down and he regularly got into (sometimes serious) trouble with local law enforcement. I visited him and picked him up from the county jail on several occasions over the years. As I got to know Jim better, I also discovered he had something of an obsession with "the devil" and demons, as well angels and manual copying of long portions of the Christian Bible.
One evening, Jim seemed to be struggling with an intense episode of paranoia and I went to visit him at his request. Upon entering his apartment, I felt that something was... wrong, very wrong. I couldn't quite figure out what it was, but then Jim spontaneously began to verbally relate knowledge of events in my life which were in no way recorded in digital or physical media (Facebook, paper trash, etc.) – private things, secret things, spanning years before I met Jim (and in precise form) of which I had never spoken. It began with one sudden statement, which startled me, which reaction elicited a pained laugh from Jim. He then proceeded to relate more "secret knowledge". His intense gaze (at me) and manner of speech throughout seemed "detached" from the world around him... for just a moment I was genuinely frightened. I said a silent prayer, abruptly changed the subject, and then hurried him out the door so we could go get a late dinner and coffee. Later conversations with Jim suggested that he didn't recall the incident; I've never asked him how he knew those things. Sadly, Jim disappared some time later, after a failed intervention to get him medical help. If he's still alive, I hope a couple of things for Jim. One, that he will realize he needs medical help to deal with his mental illness, and will avail himself of it. Two, that if he is under demonic influence, then he will meet with a kind soul who is able to help deliver him. I have spoken with a real-life exorcist – I didn't seek him out, I met him in passing – and was told that my experience with Jim was a possible sign of demonic activity though not necessarily possession.
Please note that my relating the above is not my way of "pounding the table" to insist that demons are real. It's a very personal and unexaggerated anecdote offered, for what it's worth, in relation to the idea that exorcisms and experience of the demonic would somehow always relate to "internal" phenomena.